A025367 -id:A025367 - cp's OEIS frontend

A024796 Numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 <= i <= j <= k.

27, 33, 38, 41, 51, 54, 57, 59, 62, 66, 69, 74, 75, 77, 81, 83, 86, 89, 90, 94, 98, 99, 101, 102, 105, 107, 108, 110, 113, 114, 117, 118, 121, 122, 123, 125, 126, 129, 131, 132, 134, 137, 138, 139, 141, 146, 147, 149, 150, 152, 153, 154, 155, 158, 161, 162, 164, 165, 166, 170

Offset: 1

Clark Kimberling

nonn

a(n) multiplied by (h^2)/(8*m*a^2) is the n-th energy level exhibiting accidental degeneracy, for a quantum mechanical particle of mass m in a cubic box of side length a (h is Planck's constant). - A. Timothy Royappa, Feb 12 2019

Robert Price, Table of n, a(n) for n = 1..8057

Cf. A000408, A007692, A008917, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025367, A025427.

okQ[n_]:= Length[Select[PowersRepresentations[n, 3, 2], !MemberQ[#, 0] &]] > 1; (* Jinyuan Wang, Feb 12 2019 *)

is(n)=if(n<27, return(0)); if(n%4==0, return(is(n/4))); my(w); for(i=sqrtint((n-1)\3)+1,sqrtint(n-2), my(t=n-i^2); for(j=sqrtint((t-1)\2)+1,min(sqrtint(t-1),i), if(issquare(t-j^2), w++>1 && return(1)))); 0 \\ Charles R Greathouse IV, Aug 05 2024

{n: A025427(n) > 1 }. - R. J. Mathar, Aug 05 2022

A309763 Numbers that are the sum of 4 nonzero 4th powers in more than one way.

Original entry on oeis.org

259, 2674, 2689, 2754, 2929, 3298, 3969, 4144, 4209, 5074, 6579, 6594, 6659, 6769, 6834, 7203, 7874, 8194, 8979, 9154, 9234, 10113, 10674, 11298, 12673, 12913, 13139, 14674, 14689, 14754, 16563, 16578, 16643, 16818, 17187, 17234, 17299, 17314, 17858, 18963, 19699

Offset: 1

Ilya Gutkovskiy, Aug 15 2019

nonn

			259 = 1^4 + 1^4 + 1^4 + 4^4 = 2^4 + 3^4 + 3^4 + 3^4, so 259 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A000583, A003338, A018786, A025367, A025406, A309762, A344193, A344238, A344241, A344644.

N:= 10^5: # for terms <= N
V:= Vector(N, datatype=integer[4]):
for a from 1 while a^4 <= N do
  for b from 1 to a while a^4+b^4 <= N do
    for c from 1 to b while a^4 + b^4+ c^4 <= N do
      for d from 1 to c do
         v:= a^4+b^4+c^4+d^4;
         if v > N then break fi;
         V[v]:= V[v]+1
od od od od:
select(i -> V[i]>1, [$1..N]); # Robert Israel, Oct 07 2019

Select[Range@20000, Length@Select[PowersRepresentations[#, 4, 4], ! MemberQ[#, 0] &] > 1 &]

A025406 Numbers that are the sum of 4 positive cubes in 2 or more ways.

Original entry on oeis.org

219, 252, 259, 278, 315, 376, 467, 522, 594, 702, 758, 763, 765, 802, 809, 819, 856, 864, 945, 980, 1010, 1017, 1036, 1043, 1073, 1078, 1081, 1118, 1134, 1160, 1225, 1251, 1352, 1367, 1368, 1374, 1375, 1393, 1397, 1423, 1430, 1458, 1460, 1465, 1467, 1484

Offset: 1

David W. Wilson

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003327, A008917, A025367, A025404, A025407, A025457, A309763, A343702.

N:= 2000: # for terms <= N
S2:= {}: S1:= {}:
for x from 1 while x^3 < N do
for y from 1 to x while x^3 + y^3 < N do
  for z from 1 to y while x^3 + y^3 + z^3 < N do
    for w from 1 to z do
    v:= x^3 + y^3 + z^3 + w^3;
    if v > N then break fi;
    if member(v,S1) then S2:= S2 union {v}
    else S1:= S1 union {v}
    fi
od od od od:
sort(convert(S2,list)); # Robert Israel, Feb 24 2021

{n: A025457(n) >= 2}. - R. J. Mathar, Jun 15 2018

A025368 Numbers that are the sum of 4 nonzero squares in 3 or more ways.

Original entry on oeis.org

28, 42, 52, 55, 58, 60, 63, 66, 67, 70, 73, 75, 76, 78, 79, 82, 84, 85, 87, 90, 91, 92, 93, 95, 97, 98, 99, 100, 102, 103, 105, 106, 108, 109, 110, 111, 112, 114, 115, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, 129, 130, 132, 133, 134, 135, 137, 138, 139, 140, 141

Offset: 1

David W. Wilson

nonn

Index entries for sequences related to sums of squares

Cf. A025331, A025359, A025367, A025369, A025407, A344796.

{n: A025428(n) >=3}. Union of A025369 and A025359.- R. J. Mathar, Jun 15 2018

A344795 Numbers that are the sum of five squares in two or more ways.

Original entry on oeis.org

20, 29, 32, 35, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101

Offset: 1

Sean A. Irvine, May 28 2021

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..20000

Cf. A025367, A047700, A294736, A343702, A344796, A344806.

A025358 Numbers that are the sum of 4 nonzero squares in exactly 2 ways.

Original entry on oeis.org

31, 34, 36, 37, 39, 43, 45, 47, 49, 50, 54, 57, 61, 68, 69, 71, 74, 77, 81, 83, 86, 94, 107, 113, 116, 131, 136, 144, 149, 200, 216, 272, 296, 344, 376, 464, 544, 576, 800, 864, 1088, 1184, 1376, 1504, 1856, 2176, 2304, 3200, 3456, 4352, 4736, 5504, 6016, 7424

Offset: 1

David W. Wilson

nonn

Conjecture: the even members of this sequence are all numbers of the form

k*4^m for k in [9,17,29], m>= 1, or k*4^m for k in [34, 50, 54, 74, 86, 94], m>=0. - Robert Israel, Nov 03 2017

Alois P. Heinz, Table of n, a(n) for n = 1..77 (first 71 terms from Robert Price)
Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Cf. A025367 (at least 2 ways).

N:= 10000: # to get all terms <= N
T:= Vector(N):
for a from 1 to floor(sqrt(N/4)) do
    for b from a to floor(sqrt((N-a^2)/3)) do
      for c from b to floor(sqrt((N-a^2-b^2)/2)) do
        for d from c to floor(sqrt(N-a^2-b^2-c^2)) do
          m:= a^2+b^2+c^2+d^2;
          T[m]:= T[m]+1;
od od od od:
select(i -> T[i] = 2, [$1..N]); # Robert Israel, Nov 03 2017

M = 1000;
Clear[T]; T[_] = 0;
For[a = 1, a <= Floor[Sqrt[M/4]], a++,
  For[b = a, b <= Floor[Sqrt[(M - a^2)/3]], b++,
    For[c = b, c <= Floor[Sqrt[(M - a^2 - b^2)/2]], c++,
      For[d = c, d <= Floor[Sqrt[M - a^2 - b^2 - c^2]], d++,
        m = a^2 + b^2 + c^2 + d^2;
        T[m] = T[m] + 1;
]]]];
Select[Range[M], T[#] == 2&] (* Jean-François Alcover, Mar 22 2019, after Robert Israel *)

{n: A025428(n) = 2}. - R. J. Mathar, Jun 15 2018

A047698 Numbers that are the sum of four positive seventh powers in two or more ways.

Original entry on oeis.org

2056364173794800, 12191487610289536, 263214614245734400, 696885239160606459, 1560510414117060608, 4497268448089227600, 18896044524917750533, 25704745581139341318, 26662783403703215232, 33691470623454003200, 44053472466326057527, 100502405478837434259

Offset: 1

N. J. A. Sloane

			4497268448089227600 = 30^7 + 42^7 + 369^7 + 447^7 = 45^7 + 270^7 + 387^7 + 438^7. - _Sean A. Irvine_, May 15 2021

D. J. Bernstein, Enumerating solutions to p(a) + q(b) = r(c) + s(d)
R. L. Ekl, New Results in Equal Sums of Like Powers, Math. Comput. 67, 1309-1315, 1998.

Cf. A025367, A025406, A309763.

Corrected by D. J. Bernstein (djb(AT)cr.yp.to)

a(6)-a(12) from Sean A. Irvine, May 14 2021

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A024796 Numbers expressible in more than one way as i^2 + j^2 + k^2, where 1 <= i <= j <= k.

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A309763 Numbers that are the sum of 4 nonzero 4th powers in more than one way.

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A025406 Numbers that are the sum of 4 positive cubes in 2 or more ways.

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A025368 Numbers that are the sum of 4 nonzero squares in 3 or more ways.

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A344795 Numbers that are the sum of five squares in two or more ways.

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A025358 Numbers that are the sum of 4 nonzero squares in exactly 2 ways.

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A047698 Numbers that are the sum of four positive seventh powers in two or more ways.

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